Mapping the Berry curvature from semiclassical dynamics in optical lattices

We propose a general method by which experiments on ultracold gases can be used to determine the topological properties of the energy bands of optical lattices, as represented by the map of the Berry curvature across the Brillouin zone. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wave packet undergoing Bloch oscillations. However, in two dimensions these trajectories may be complicated Lissajous-like figures, making it difficult to extract the effects of Berry curvature in general. We propose how this can be done using a “time-reversal” protocol. This compares the velocity of a wave packet under positive and negative external force, and allows a clean measurement of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three specific systems: the asymmetric hexagonal lattice and two “optical flux” lattices in which the Chern number is nonzero. Finally, we discuss general experimental considerations for observing Berry curvature effects in ultracold gases.

Figure 1

An example of a Lissajous-like figure for the square optical flux lattice (Sec. 5b1). The ratio $F_x:F_y$ is $9:1$0, and a low force of $\left|\mathbf{F}\right|=0.05F_R$ is used to minimize the effects of Berry curvature over one oscillation ($F_R=h^2/2m{\lambda }^3$, where $\lambda$ is the optical wavelength and $m$ is the mass of the atomic species; see Sec. 5). The Lissajous-like figure is approximately bounded by the Bloch oscillation lengths, and so it obscures the effects of Berry curvature within this box.

Figure 2

Two methods for mapping the Brillouin zone. (a) Rotating the force with respect to the lattice between experiments allows each wave packet to travel a different path. (b) With a large ratio $F_x:F_y$, a single wave packet successively travels many paths within the Brillouin zone.

Figure 3

The asymmetric hexagonal lattice in real space (on the left) and reciprocal space (on the right). The real-space lattice vectors are ${\mathbf{a}}_1=a(\sqrt{3}/2,-1/2)$ and ${\mathbf{a}}_2=a(0,1)$, where $a$ is the lattice spacing (and for this geometry, $a=2\lambda /3$). The sublattices are connected by ${\mathbf{R}}_1=a(1/2\sqrt{3},1/2)$, ${\mathbf{R}}_2=a(1/2\sqrt{3},-1/2)$, and ${\mathbf{R}}_3=a(-1/\sqrt{3},0)$. The dotted lines indicate the unit cell. The reciprocal lattice vectors are then ${\mathbf{K}}_1=(4\pi /\sqrt{3}a)(1,0)$ and ${\mathbf{K}}_2=(4\pi /\sqrt{3}a)(1/2,\sqrt{3}/2)$.

Figure 4

(a) Band structure of the lowest band of the asymmetric hexagonal lattice for $W=0.5E_R$ and $t=1.0E_R$, and the energy in units of $E_R$ (for this geometry, $a=2\lambda /3$). Due to the asymmetry, gaps have opened at the Dirac points at the corners of the Brillouin zone. (b) The Berry curvature mapped out for $W=0.5E_R$, using the method of Ref. [45]. Light shading indicates $\Omega >0$ and dark shading indicates $\Omega <0$. Positive and negative regions cancel, giving a net Chern number of zero.

Figure 5

(a) The trajectory of a semiclassical wave packet through the Brillouin zone, starting from $\mathbf{k}=(0,0)$ with $\mathbf{F}=(0.0,1.0F_R)$. The numbers indicate the order in which the path is traveled. (b) The corresponding real-space trajectory of the wave packet, starting from the origin, for $W=0.5E_R$. This result was previously obtained in Ref. [33]. The motion along $x$ is due to the Berry curvature, while that along $y$ is due to the band structure.

Figure 6

(a) Velocity along $x$ of a wave packet moving under the force $\mathbf{F}=\pm (0.0,1.0F_R)$ with $W=0.5E_R$. $k_y$ is measured along the path traveled (which is periodic in $4\pi$). (b) Velocity along $y$. (c) Applying the time-reversal protocol to extract the Berry velocity: $\mathbf{v}(+\mathbf{F})-\mathbf{v}(-\mathbf{F})=-2/\hslash (\mathbf{F}\times \widehat{\mathbf{z}})\Omega \left(\mathbf{k}\right)$. (d) Applying the time-reversal protocol to extract the group velocity: $\mathbf{v}(+\mathbf{F})+\mathbf{v}(-\mathbf{F})=(2/\hslash )\partial \varepsilon \left(\mathbf{k}\right)/\partial \mathbf{k}$.

Figure 7

Band structure of the lowest band in the magnetic Brillouin zone for the one-photon square optical flux lattice with (a) $V=0.0E_R$, (b) $V=1.0E_R$, and (c) $V=3.0E_R$, and the energy in units of $E_R$. At $V=0.0E_R$, the lowest band touches the second lowest band at $k=(\pi /a,\pm \pi /a)$. As $V$ is turned on, band gaps open at these points and Berry curvature is formed.

Figure 8

Contour maps of the Berry curvature, $\Omega \left(\mathbf{k}\right)$, for the square optical flux lattice when (a) $V=1.0E_R$ and (b) $V=3.0E_R$. $\Omega$ is in units of $1/\text{(MBZ}\text{area)}=a^2/8{\pi }^2$. $\Omega$ is concentrated at points $k=(\pi /a,\pm \pi /a)$, spreading out with increasing $V$.

Figure 9

(a) The trajectory of a semiclassical wave packet through the magnetic Brillouin zone, starting from $\mathbf{k}=(0,0)$ with $\left|\mathbf{F}\right|=1.0F_R$ directed along (1,1). The numbers indicate the order in which the path is traveled. The dotted line indicates the simple Brillouin zone, while the dashed lines show the extension into the magnetic Brillouin zone. (b) The corresponding real-space trajectory of the wave packet starting from the origin, for $V=0.4E_R$ and $V=3.0E_R$. The motion perpendicular to $\mathbf{F}$ is due to the Berry curvature, while that parallel is due to the band structure.

Figure 10

The velocity of the wave packet (a) perpendicular and (b) parallel to the applied force for $\left|\mathbf{F}\right|=1.0F_R$ directed along (1,1) and $V=0.4E_R$. $k$ is measured along the path traveled. For this simple alignment of the force, (a) contains the effects of Berry curvature along the path, while (b) shows the group velocity.

Figure 11

Trajectory for a wave packet traveling with (a) $\left|\mathbf{F}\right|=1.0F_R$ and (b) $\left|\mathbf{F}\right|=5.0F_R$ where the force is aligned such that $F_x:F_y=1:16$ and $V=0.4E_R$. The Bloch motion is no longer purely along the direction of the force, and the resultant trajectory is complex. The net drift between Bloch oscillations is a measure of the total Berry curvature along a path, but other information is obscured. The trajectory due to the group velocity increases in size relative to the trajectory due to the Berry curvature with decreasing force (Sec. 6), leading to more complicated motion in (a) than (b).

Figure 12

(a) Velocity along $x$ of a wave packet moving under the force $\left|\mathbf{F}\right|=1.0F_R$ with $F_x:F_y=1:16$ and $V=0.4E_R$. $k$ is measured along the path traveled. (b) Velocity along $y$. (c) Applying the time-reversal protocol to extract the Berry velocity. The Berry velocity calculated from $v_x$ and $v_y$ differ by a factor of 16 from the ratio of the forces, and by a sign, due to the cross product. (d) Applying the time-reversal protocol to extract the group velocity. As expected, there are 16 oscillations from $v_y$ for every one from $v_x$.

Figure 13

As in Fig. 12 but now for $\left|\mathbf{F}\right|=5.0F_R$.

Figure 14

$|v_x(+\mathbf{F})-v_x(-\mathbf{F})|/2=1/\hslash \left(F_y\Omega \right)$ plotted along the path taken by the wave packet in the Brillouin zone for $\left|\mathbf{F}\right|=1.0F_R$ with $F_x:F_y=1:16$ and $V=0.4E_R$. This technique will enable experiments to directly map out the Berry curvature.

Figure 15

(a) The energy (measured in $E_R$) of the lowest band for $V=1.8E_R$, $\theta =0.3$, and $\epsilon =0.4$. For these parameters, the lowest band has a Chern number of one. (b) The corresponding Berry curvature. $\Omega \left(\mathbf{k}\right)$ is in units of $1/\text{(BZ}\text{area)}$. In this case, the Berry curvature is significant over much of the Brillouin zone and has regions of both positive and negative sign.

Figure 16

The real-space trajectory of a wave packet under a force $\mathbf{F}=(0.0,0.1F_R)$ over one period, for $V=1.8E_R$, $\theta =0.3$, and $\epsilon =0.4$. The wave packet follows the path described in Fig. 5. While the group velocity is again purely along $y$, the motion due to the Berry curvature is more complicated due to the varying sign of $\Omega$.

Figure 17

(a) Oscillation in expectation value of ${\widehat{\sigma }}_x$ for the one-photon optical flux lattice, as the wave packet travels along $(1,1)$. The dashed line is for $V=0.04E_R$, while the smooth line is $V=0.4E_R$. The area over which the transfer occurs decreases with potential in a similar way to how the Berry curvature area decreases. $k$ is measured along the path of the wave packet. (b) Oscillation in population in the two states for the two-photon optical flux lattice as the wave packet travels along $(0,1)$ for $V=1.8E_R$, $\theta =0.3$, and $\epsilon =0.4$. The maximum population imbalance appears to be around points $K$ and $K^{\prime }$. $k_y$ is measured along the path of the wave packet.